I dunno what you would hope to gain by making the .308 Winchester an "improved" cartridge. It's alrealy close to what most "improved cartridges" are: short neck with sharp shoulder, minimum body taper, and close to optimum powder capacity for it's bore size.
The .308 W is not chambered in Contenders for a simple reason: the gun won't hold up to extensive firing, and will result in the lockup loosening, stretched frames, and other bad things. This is not a function of it being a tapered case, but because of the high SAAMI pressures it's loaded to, and the large .473" case give lots of back-thrust on the breech.
Other big cartridges use large, minimally tapered cases too: .444 Remington Magnum, .454 Casull, .45 Winchester Magnum, .300 Savage, and so on. But the .308 Winchester has the highest standard SAAMI pressure, and combined with the .473" case head, and enthusiastic handloaders, can easily lead to BIG PROBLEMS.